9.3 Toric morphisms and subdivisions

Toric morphisms are maps between toric varieties that respect their torus actions. They come from maps between fans, which are collections of cones describing the varieties. Fan subdivisions refine these cones, creating new toric morphisms.

These ideas connect toric varieties to polyhedral geometry, a key theme of this unit. By studying toric morphisms and fan subdivisions, we can understand relationships between different toric varieties and how they transform.

Toric Morphisms and Fan Subdivisions

Definition and Relationship

• A toric morphism is a morphism between two toric varieties that is equivariant with respect to the torus action on each variety
• Toric morphisms arise from morphisms of fans, which are maps between the lattices of the fans compatible with the fan structure
• A fan subdivision refines a fan by subdividing its cones into smaller cones while preserving the fan structure and the lattice
• Fan subdivisions induce toric morphisms between the corresponding toric varieties, which are proper and birational morphisms

Examples

• Consider the fan of $\mathbb{P}^2$ and its subdivision obtained by adding a ray through the midpoint of a cone. This subdivision induces a toric morphism from the blowup of $\mathbb{P}^2$ at a torus-fixed point to $\mathbb{P}^2$
• The identity map on a fan induces the identity toric morphism on the corresponding toric variety

Constructing Toric Morphisms

Construction from Fan Subdivisions

• Given a fan subdivision, the corresponding toric morphism is constructed by defining a map between the toric varieties induced by the map between the fans
• The toric morphism can be described explicitly in terms of the homogeneous coordinates of the toric varieties using the combinatorial data of the fan subdivision
• Toric morphisms are determined by their action on the torus-invariant affine open subsets of the toric varieties, which correspond to the cones of the fans

Examples

• Let $\Sigma$ be the fan of $\mathbb{P}^2$ and $\Sigma'$ its subdivision obtained by adding a ray through a point on the boundary of a cone. The toric morphism induced by this subdivision maps the exceptional divisor of the blowup to the torus-invariant point corresponding to the subdivided cone
• The toric morphism induced by a star subdivision of a fan is given by the inclusion of the torus-invariant open subset corresponding to the subdivided cone into the original toric variety

Properties of Toric Morphisms

Isomorphisms and Trivial Subdivisions

• A toric morphism is an isomorphism if and only if the fan subdivision is trivial, i.e., the identity map on the fan
• Trivial fan subdivisions induce toric isomorphisms between the same toric variety

Properness and Refinements

• A toric morphism is proper if and only if the fan subdivision is a refinement, i.e., every cone of the codomain fan is subdivided into cones of the domain fan
• Proper toric morphisms are induced by fan refinements, which increase the level of detail in the fan structure

Birationality and Star Subdivisions

• A toric morphism is birational if and only if the fan subdivision is a star subdivision, i.e., only the top-dimensional cones are subdivided, and the lattice is preserved
• Star subdivisions induce birational toric morphisms, which are isomorphisms outside the exceptional locus

Fibers and Preimages of Cones

• The fiber of a toric morphism over a point in the codomain toric variety can be described in terms of the preimage of the corresponding cone under the fan subdivision
• The preimage of a cone under a fan subdivision determines the structure of the fiber over the corresponding torus-invariant point

Examples

• The blowup of $\mathbb{P}^2$ at a torus-fixed point is a proper toric morphism induced by a refinement of the fan of $\mathbb{P}^2$
• The blowup of $\mathbb{P}^2$ at a point is a birational toric morphism induced by a star subdivision of the fan of $\mathbb{P}^2$
• The fiber of the blowup of $\mathbb{P}^2$ at a torus-fixed point over that point is the exceptional divisor, which is a $\mathbb{P}^1$ corresponding to the preimage of the subdivided cone

Toric Morphisms for Variety Relationships

Birational Equivalence

• Two toric varieties are birationally equivalent if and only if their fans are related by a sequence of star subdivisions and their inverses
• Birational equivalence of toric varieties can be studied using toric morphisms induced by star subdivisions

Deformations

• A toric variety can be deformed into another toric variety by a sequence of toric morphisms induced by fan subdivisions that preserve the support of the fan
• Deformations of toric varieties correspond to certain fan subdivisions that maintain the overall structure of the fan

Degenerations

• A toric variety can degenerate into another toric variety by a toric morphism induced by a fan subdivision that maps a higher-dimensional cone onto a lower-dimensional cone
• Degenerations of toric varieties are induced by fan subdivisions that collapse some cones to lower dimensions

Resolutions of Singularities

• Toric morphisms can be used to construct resolutions of singularities of toric varieties by subdividing the fan until all the cones are smooth
• Resolving singularities of toric varieties amounts to finding suitable fan subdivisions that eliminate the singular cones

Examples

• $\mathbb{P}^2$ and the Hirzebruch surface $\mathbb{F}_1$ are birationally equivalent toric varieties related by a star subdivision of the fan of $\mathbb{P}^2$
• The family of Hirzebruch surfaces $\mathbb{F}_n$ can be obtained by deforming $\mathbb{F}_1$ using toric morphisms induced by fan subdivisions that preserve the support of the fan
• The weighted projective space $\mathbb{P}(1,1,2)$ degenerates to the projective cone over a conic by a toric morphism induced by a fan subdivision that maps the cone corresponding to the singular point onto the ray corresponding to the cone point
• The singularity of the affine cone over a conic can be resolved by a toric morphism induced by a fan subdivision that adds rays to subdivide the singular cone into smooth cones